Pablo Monzón

Global considerations on the Kuramoto model of sinusoidally coupled oscillators

In this article we study global stability properties of the Kuramoto model of sinusoidally coupled oscillators. We base our analysis on previous results by the control community that analyze local properties of the consensus point of different kinds of Kuramoto models. We prove that for the complete symmetric case, the consensus point is almost globally stable, that is, the set of trajectories that do not converge to it has zero measure. We present a counter-example of that when the completeness hypothesis is removed. We also show that the general non-symmetric case is more complex and we analyze the particular case of oscillators coupled in a ring structure, where we can establish some global stability properties.

Almost Global Synchronization of Symmetric Kuramoto Coupled Oscillators

A few decades ago, Y. Kuramoto introduced a mathematical model of weakly coupled oscillators that gave a formal framework to some of the works of A.T. Winfree on biological clocks [Kuramoto (1975), Kuramoto (1984), Winfree (1980)]. The model proposes the idea that several oscillators can interact in a way such that the individual oscillation properties change in order to achieve a global behavior for the interconnected system. The Kuramoto model serves as a good representation of many systems in several contexts: biology, engineering, physics, mechanics, etc. [Ermentrout (1985), York (1993), Strogatz (1994), Dussopt et al. (1999), Strogatz (2000), Jadbabaie et a. (2003), Rogge et al. (2004), Marshall et al. (2004), Moshtagh et al. (2005)]. Recently, many works on the control community have focused on the analysis of the Kuramoto model, specially the one with sinusoidal coupling. The consensus or collective synchronization of the individuals is particularly important in many applications representing coordination, cooperation, emerging behavior, etc. Local stability properties of the consensus have been initially explored in [Jadbabaie et al. (2004)]. It must be noted that little attention has been devoted to the influence of the underlying interconnection graph on the stability properties of the system. The reason could be the fact that the local stability does not depend on the interconnection [van Hemmen et al. (1993)]. Global or almost global dynamical properties were studied in [Monzon et al. (2005), Monzon (2006), Monzon et al. (2006)]. In these works, the relevance of the interconnection graph of the system was hinted. In the present chapter, we go deeper on the analysis of the relationships between the dynamical properties of the system and the algebraic properties of the interconnection graph, exploiting the strong algebraic structure that every graph has. We step forward into a classification of the interconnection graphs that ensure almost global attraction of the set of synchronized states. In Section 2 we present the Kuramoto model for sinusoidally coupled oscillators, its general properties and the notion of almost global synchronization; in Section 3 we review some basic facts on algebraic graph theory; the symmetric Kuramoto model and the block analysis are presented in Sections 4 and 5; Section 6 gives some examples and applications of the main results; Section 7 presents the problem of classification of almost global synchronizing topologies.

Gluing Kuramoto coupled oscillators networks

In this work we prove that the problem of almost global synchronization of the Kuramoto model of sinusoidally symmetric coupled oscillators with a given topology could be reduced to the analysis of the blocks of the underlying interconnection graph.

Modal analysis of the Uruguayan electrical power system

The Uruguayan electric power system is facing deep challenges in the near future. The plans for 2015 include a significant expansion of generation based on renewable sources (wind, biomass) and a new HVDC international interconnection. These plans and the opening to multiple, distributed private generators pose significant challenges to the system operation and planning. This work describes the modal analysis of the Uruguayan Power System by considering a base 2004 scenario and two 2010 scenarios. These scenarios include conventional-thermal and hydro synchronous generation-and constitute a reference for a set of ongoing studies associated with the new expansion plans. The system comprises a relatively small electrical network with important hydroelectrical generation and highly interconnected with the neighboring countries. The main oscillation modes, which exhibit a poorly damped behavior, are analyzed in detail. The results were validated through simulations of the systemss response for different contingencies. The study includes the selection of appropriate machines to damp the oscillations and the tuning of Power System Stabilizers, PSS. The effects of this control action are then assessed via the modal analysis and validated via complete non linear simulation of transient response.

Local and global aspects of almost global stability

In this work we introduce several known and new results on almost global stability. We focus on how local properties of equilibrium points of dynamical systems are related to the existence of density functions and to the almost global stability property. We present some examples that illustrate those results

Local implications of almost global stability

We prove that for autonomous differential equations with the origin as a fixed point, and with at least one eigenvalue with negative real part, we have that the existence of a density function implies local asymptotical stability. We present examples of a system admitting a density function for which the origin is not locally asymptotically stable and an almost globally stable system for which no function is a density function.

Global Synchronization Properties for Different Classes of Underlying Interconnection Graphs for Kuramoto Coupled Oscillators

This article deals with the general ideas of almost global synchronization of Kuramoto coupled oscillators and synchronizing graphs. We review the main existing results and introduce new results for some classes of graphs that are important in network optimization: complete k -partite graphs and what we have called Monma graphs.

On the Characterization of Families of Synchronizing Graphs for Kuramoto Coupled Oscillators

Abstract Kuramoto model of coupled oscillators represents situations where several individual agents interact and reach a collective behavior. The interaction is naturally described by a interconnection graph. Frequently, the desired performance is the synchronization of all the agents. Almost global synchronization means that the desire objective is reached for every initial conditions, with the possible exception of a zero Lebesgue measure set. This is a useful concept, specially when global synchronization can not be stated, due, for example, to the existence of multiple equilibria. In this survey article, we give an analysis of the influence of the interconnection graph on this dynamical property. We present in a ordered way several known and new results that help on the characterization of what we have called synchronizing topologies.

Decision making in forward power markets with supply and demand uncertainty

The paper studies forward markets of electric energy, where generators and consumers bid for quantities of energy ahead of time, but face uncertainty on their real-time supply or demand, expressed through a probability distribution. The optimal forward decision is derived in both cases, providing extensions to the recent literature on this topic. In particular, the case where demand is elastic in addition to uncertain is addressed in detail. Finally, we analyze the integrated forward market where buyers and sellers of random energy interact with dispatchable sellers to determine a clearing price.

Almost global attraction in planar systems

Abstract In this work, we present a result relating the recent ideas of almost global stability and density functions with the classical Poincare–Bendixson theory for planar systems.